Abstract
Let \(\mathcal{F}\mathcal{B}_{n}(\Omega )\) denote operators in the Cowen–Douglas class \(\mathcal {B}_{n}(\Omega )\) possessing a flag structure. All the irreducible homogeneous operators in \(\mathcal {B}_{n}(\Omega )\) belong to this class. The unitary invariants of this class of operators include the curvature and the second fundamental form of the corresponding line bundle. In this paper, we introduce a subclass of \(\mathcal{F}\mathcal{B}_{n}(\Omega )\) which possesses a “strong" flag structure, and for which the curvature and the second fundamental form of the associated line bundle is a complete set of unitary invariants. We prove that this new class of operators is norm dense in \(\mathcal {B}_{n}(\Omega )\) up to similarity. We obtain a classification modulo conjugation by an invertible operator for a large class of operators possessing a strong flag structure. Along the way, it is shown that the number of the similarity invariants found recently can be reduced from \(\frac{n(n-1)}{2}+1\) to n. Moreover, we obtain a complete characterization of weakly homogeneous operators with large index and flag structure.
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Acknowledgements
The authors thank the referees for several constructive suggestions. This has greatly helped us in preparing a more complete and final version of the manuscript. The second author is supported by National Natural Science Foundation of China (Grant No. 12371129 and 11920101001).
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**e, Y., Ji, K. Operators of the Cowen–Douglas class with strong flag structure. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 132 (2024). https://doi.org/10.1007/s13398-024-01630-y
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DOI: https://doi.org/10.1007/s13398-024-01630-y